Functors for Long dimodules and Yetter–Drinfeld modules in a weak setting

Abstract

In this paper, for two weak Hopf monoids $H$ and $B$ with invertible antipode, we define a functor between the category of left-left $H\otimes B$-Yetter–Drinfeld modules and the category of $H$-$B$-Long dimodules. We also show that, if moreover $H$ is quasitriangular and $B$ is coquasitriangular, this functor is a retraction of the well-known injective functor between left-left $H$-$B$-Long dimodules and left-left $H\otimes B$-Yetter–Drinfeld modules.